understanding the "Offering RSA public key" step during SSH connection initialization

When the SSH client displays this message, it's trying to authenticate the user on the server (userauth_pubkey in sshconnect2.c). The client needs to demonstrate that it has the private key corresponding to a public key that is authorized on the server. The file name displayed in the debug message is the name of the private key file (e.g. passed as an argument to -i or as the IdentityFile configuration directive).

At the point where this message is displayed, the client doesn't use the private key, only the public key. However, the client wants to know that the private key is available, because if the server agrees to use this public key then the client will have to demonstrate that it knows the private key. The client sends an SSH_MSG_USERAUTH_REQUEST message to the server with the publickey method containing the public key. If the server agrees to use this public key (“debug1: Server accepts key”) then the client will later use the private key to sign a challenge sent by the server in another SSH_MSG_USERAUTH_REQUEST message (in sign_and_send_pubkey — the have_sig byte changes from 0 (“tell me if you like this key”) to 1 (“here's a proof that I'm me, let me in”)).

My guess would be that the client announce the key pair he's using, giving the public key (of course!). But indeed, the key given with that message is a private key, which I don't fully understand why. I tried with a ssh -vvv with a server I use:

debug1: Offering RSA public key: /home/user/.ssh/id_rsa
debug3: send_pubkey_test
debug2: we sent a publickey packet, wait for reply
debug1: Authentications that can continue: publickey,gssapi-keyex,gssapi-with-mic,password
debug1: Trying private key: /home/user/.ssh/id_dsa
debug3: no such identity: /home/user/.ssh/id_dsa: No such file or directory
debug1: Trying private key: /home/user/.ssh/id_ecdsa
debug3: no such identity: /home/user/.ssh/id_ecdsa: No such file or directory
debug1: Offering ED25519 public key: /home/user/.ssh/id_ed25519
debug3: send_pubkey_test
debug2: we sent a publickey packet, wait for reply
debug1: Server accepts key: pkalg ssh-ed25519 blen 51

As I have 2 keys on this computer: the RSA one and the ECDSA one, it looks like the server rejected my RSA key. That's ok, I did not provide the RSA key to this server. Then, he tries to find a DSA and a ECDSA key, but I haven't any of these, so the "no such identity" is understandable. Then, I own a ED25519 key, and it tries this one. And that goes well, since I registered this key pair to that server. Note that it accepts my public key now!

Then, the protocol keeps running through a challenge where the server now knows which public key use to cypher the challenge, and the client (my machine) knows which private key to use for this challenge.